A sudden thought occured
Try changing this line:
if (!card.init(SPI_HALF_SPEED, chipSelect)) {
To:
if (!card.init(6, chipSelect)) {
That will slow down the SPI to the slowest possible speed of FCPU/128
If that works, try decreasing the number to 5, then 4, then 3 and so on. Each time it will double the speed.
With simple level converters like these, I was never able to get it to run faster than FCPU/8 (aka SPI_QUARTER_SPEED) reliably. Once i switched to a proper level shifter like the TXB0104, I could get up to the full speed of FCPU/2 (aka SPI_FULL_SPEED).
EDIT:
Just as a side note, the card.init() call initialises the card at FCPU/128, and then at the end of the routine sets it to full speed. This means that the volume.init() call will be running on a an SPI at a much higher freqency than card.init() which could be the reason why it fails at that point.